ORBITS IN FREE ALGEBRAS OF RANK-2

Let F be an algebra which is free in a class of algebras. In this pape r we consider the following problem: If an endomorphism psi psi of F p reserves the orbit (AutF)(omega) of a nontrivial element omega is an e lement of F, does this imply that psi is an automorphism? We give posi tive solutions of this problem when F is generated by two elements and is the free Lie algebra, free metabelian Lie algebra or free metabeli an associative algebra, and omega ia certain kind of element.